Help Homework : MaKE A TABLE:relate primeter and area
Amaya has 24 inches of ribbon she can glue around the edge of a rectangular gift box lid. if she wants to use all the ribbon without any overlapping, what are the length and width of a gift box lid with the greatest area?
area= lw
but 24=2L+2w or w=12-L
area=l(12-l)
Wondering if you have calculus yet....assuming not.
table
l area
4 32
5 35
6 36
7 35...already found max area, l=6, so w=6
Length = 7 inches
Width = 5 inches
Area = 35 square inches
Amaya has 24 inches of ribbon she can glue around the edge of a rectangular gift box lid. if she wants to use all the ribbon without any overlapping, what are the length and width of a gift box lid with the greatest area?
To create a table relating the perimeter and area of a rectangle, we need to understand the formulas for calculating the perimeter and area first.
1. Perimeter: The perimeter of a rectangle is the total length of all its sides. For a rectangle, we can calculate the perimeter by using the formula:
P = 2(l + w)
where P represents the perimeter, l represents the length, and w represents the width.
2. Area: The area of a rectangle is the measure of the region enclosed by the rectangle. For a rectangle, we can calculate the area by using the formula:
A = l * w
where A represents the area, l represents the length, and w represents the width.
Now, let's create a table to relate the perimeter and area:
Perimeter (P) | Length (l) | Width (w) | Area (A)
------------------------------------------------
| | | |
To find the length and width of the gift box lid with the maximum area using all 24 inches of ribbon, we can use the given information:
Amaya has 24 inches of ribbon, which she can glue around the edge of the gift box lid. Without overlapping, the length of ribbon would be equal to the perimeter of the gift box lid.
Let's substitute the perimeter formula for P and rearrange the formula to find the width in terms of length:
P = 2(l + w)
24 = 2(l + w)
12 = l + w
w = 12 - l
Now, we can substitute the width formula into the area formula and simplify:
A = l * w
A = l * (12 - l)
A = 12l - l^2
To find the length and width of the gift box lid with the greatest area, we need to maximize the area formula A. We can do this by finding the critical points of the equation and determining where the area is at its maximum.
To find the critical points, we take the derivative of the area function with respect to length, set it equal to zero, and solve for length:
dA/dl = 12 - 2l
12 - 2l = 0
l = 6
Now, we know that the length (l) is 6 inches. To find the width, we can substitute this value back into the width formula:
w = 12 - l
w = 12 - 6
w = 6
So, the length and width of the gift box lid with the greatest area, using all 24 inches of ribbon, are 6 inches each.
Here's the completed table:
Perimeter (P) | Length (l) | Width (w) | Area (A)
------------------------------------------------
24 | 6 | 6 | 36
Therefore, to maximize the area using all 24 inches of ribbon, the gift box lid should have dimensions of 6 inches by 6 inches.